发表机构
Robert Bosch Center for Cyber-Physical Systems (RBCCPS); Indian Institute of Science; Cisco India; University of Melbourne; Centre for Infrastructure, Sustainable Transportation and Urban Planning (CiSTUP)（罗伯特博世网络物理系统中心（RBCCPS）; 印度科学学院; 思科印度公司; 墨尔本大学; 基础设施、可持续交通与城市规划中心（CiSTUP））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出面向双聚类的内部验证指标NVE及变体NVE₍cov₎，通过引入可分性与覆盖率考量，补充了传统仅评估一致性的双聚类验证方法，经实验验证其能更全面评估双聚类质量。
AI 中文摘要
双聚类（又称共聚类）旨在通过同时对数据矩阵的行和列进行分组，发现具有一致性的子矩阵。这种局部二维结构使得其验证比普通聚类更为困难，普通聚类的内部指标通常依赖于单一共享特征空间中的紧凑性与分离性。现有的流行双聚类内部度量，如均方残差（Mean Squared Residue, MSR）和虚拟误差（Virtual Error, VE），主要评估双聚类内部的一致性，尽管有用，但这些度量并未直接评估提取的双聚类是否相互区分，或是否解释了数据矩阵的有意义部分。本文研究了归一化虚拟误差（Normalised Virtual Error, NVE），这是一种使用超双聚类归一化策略扩展VE的内部验证指标。通过将每个双聚类的VE与其与其他双聚类合并后得到的VE进行比较，NVE引入了可分性与冗余的相对概念。我们还研究了一种覆盖率调整变体NVE₍cov₎，它会惩罚那些仅选择非常小子矩阵以获得低误差的解决方案。通过受控合成基准和酵母基因表达数据集，我们检验了NVE和NVE₍cov₎是否提供了超出标准基于一致性的度量的信息。结果表明，NVE对冗余和可分性差的双聚类敏感，而当低误差双聚类仅覆盖矩阵的可忽略部分时，NVE₍cov₎会改变解决方案的排名。这些发现表明，基于NVE的度量是内部共聚类验证的有用补充标准，尤其在必须共同考虑一致性、可分性和覆盖率时。
英文摘要
Biclustering, or co-clustering, aims to discover coherent submatrices by grouping rows and columns of a data matrix simultaneously. This local two-dimensional structure makes validation more difficult than in ordinary clustering, where internal indices usually rely on compactness and separation in a single shared feature space. Existing popular internal biclustering measures such as Mean Squared Residue (MSR), and Virtual Error (VE) mainly evaluate within-bicluster coherence. Although useful, these measures do not directly assess whether the extracted biclusters are mutually distinct or whether they explain a meaningful portion of the data matrix. This paper investigates Normalised Virtual Error (NVE), an internal validation metric that extends VE using a super-bicluster normalization strategy. By comparing the VE of each bicluster with the VE obtained after merging it with other biclusters, NVE introduces a relative notion of separability and redundancy. We also study a coverage-adjusted variant, NVE\textsubscript{cov}, which penalizes solutions that obtain low error by selecting only very small submatrices. Through controlled synthetic benchmarks and yeast gene-expression datasets, we examine whether NVE and NVE\textsubscript{cov} provide information beyond standard coherence-based metrics. The results show that NVE is sensitive to redundant and poorly separated biclusters, while NVE\textsubscript{cov} changes solution rankings when low-error biclusters cover only a negligible part of the matrix. These findings suggest that NVE-based measures are useful complementary criteria for internal co-clustering validation, especially when coherence, separability, and coverage must be considered jointly.
